For the second part of bound in (REF ), we use Assouad's method and modify the proof in [1]}. Consider \(\Sigma _1=\mathbf {I}_p\)  and \(\Sigma _2=\mathbf {I}_p+\mathbf {D}\) , where \(\mathbf {D}_{1,j}=\mathbf {D}_{j,1}=n^{-(\alpha +1)/(2\alpha +1)}\)  for \(j=1,...,n^{1/(2\alpha +1)}\) . Denote \(k=n^{1/(2\alpha +1)}\)  and \(a=n^{-(\alpha +1)/(2\alpha +1)}\) , then \(\mathbf {D}\)  is just a matrix where the first \(k\)  elements in the first row and first column are \(a\) .
